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Chapter 5: Relationships with Triangles

Introduction

This chapter introduces different segments within triangles and how they relate to each other. We will explore the properties of midsegments, perpendicular bisectors, angle bisectors, medians, and altitudes. Next, we will look at the relationship of the sides of a triangle, how they relate to each other and how the sides of one triangle can compare to another.

Chapter Summary

Summary

This chapter begins with an introduction to the Midsegment Theorem. The definition of a perpendicular bisector is presented and the Perpendicular Bisector Theorem and its converse are explored. Now that the bisectors of segments have been discussed, the definition of an angle bisector is next and the Angle Bisector Theorem and its converse are presented. The properties of medians and altitudes of triangles are discussed in detail. The entire chapter builds to a discovery of the relationships between the angles and sides in triangles as a foundation for the Triangle Inequality Theorem. The chapter ends with a presentation of indirect proofs.

Chapter Keywords

Midsegment

Midsegment Theorem

Perpendicular Bisector Theorem

Perpendicular Bisector Theorem Converse

Point of Concurrency

Circumcenter

Concurrency of Perpendicular Bisectors Theorem

Angle Bisector Theorem

Angle Bisector Theorem Converse

Incenter

Concurrency of Angle Bisectors Theorem

Median

Centroid

Concurrency of Medians Theorem

Altitude

Orthocenter

Triangle Inequality Theorem

SAS Inequality Theorem

SSS Inequality Theorem

Indirect Proof

Chapter Review

If and are the midpoints of the sides they lie on, find:

The perpendicular bisector of .

The median of .

The angle bisector of .

A midsegment.

An altitude.

Trace onto a piece of paper with the perpendicular bisector. Construct another perpendicular bisector. What is the point of concurrency called? Use this information to draw the appropriate circle.

Trace onto a piece of paper with the angle bisector. Construct another angle bisector. What is the point of concurrency called? Use this information to draw the appropriate circle.

Trace onto a piece of paper with the median. Construct another median. What is the point of concurrency called? What are its properties?

Trace onto a piece of paper with the altitude. Construct another altitude. What is the point of concurrency called? Which points of concurrency can lie outside a triangle?

A triangle has sides with length and . Find the range of the third side.

Texas Instruments Resources

In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9690.

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Description

This chapter continues to explore triangles with an emphasis on triangle relationships, properties of segments and angles, and the properties associated with the sides of triangles.